Revista de la
Unión Matemática Argentina

Volume 67, number 1 (2024)

June 2024
Front matter
Three-dimensional $C_{12}$-manifolds. Gherici Beldjilali
The present paper is devoted to three-dimensional $C_{12}$-manifolds (defined by D. Chinea and C. Gonzalez), which are never normal. We study their fundamental properties and give concrete examples. As an application, we study such structures on three-dimensional Lie groups.
1–14
On an extension of the Newton polygon test for polynomial reducibility. Brahim Boudine
Let $R$ be a commutative local principal ideal ring which is not integral, $f$ a polynomial in $R[x]$ such that $f(0) \neq 0$ and $N(f)$ its Newton polygon. If $N(f)$ contains $r$ sides of different slopes, we show that $f$ has at least $r$ different pure factors in $R[x]$. This generalizes the Newton polygon method over a ring which is not integral.
15–25
Summing the largest prime factor over integer sequences. Jean-Marie De Koninck and Rafael Jakimczuk
Given an integer $n\ge 2$, let $P(n)$ stand for its largest prime factor. We examine the behaviour of $\sum\limits_{n\le x \atop n\in A} P(n)$ in the case of two sets $A$, namely the set of $r$-free numbers and the set of $h$-full numbers.
27–35
New classes of statistical manifolds with a complex structure. Mirjana Milijević
We define new classes of statistical manifolds with a complex structure. Motivation for our work is the classification of almost Hermitian manifolds with respect to the covariant derivative of the almost complex structure, obtained by Gray and Hervella in 1980. Instead of the Levi-Civita connection, we use a statistical one and obtain eight classes of Kähler manifolds with the statistical connection. Besides, we give some properties of tensors constructed from covariant derivative of the almost complex structure with respect to the statistical connection. From the obtained properties, further investigation of statistical manifolds is possible.
37–45
Coordinate rings of some $\mathrm{SL}_2$-character varieties. Vicente Muñoz and Jesús Martín Ovejero
We determine generators of the coordinate ring of $\mathrm{SL}_2$-character varieties. In the case of the free group $F_3$ we obtain an explicit equation of the $\mathrm{SL}_2$-character variety. For free groups $F_k$, we find transcendental generators. Finally, for the case of the $2$-torus, we get an explicit equation of the $\mathrm{SL}_2$-character variety and use the description to compute their $E$-polynomials.
47–64
Spectrality of planar Moran–Sierpinski-type measures. Qian Li and Min-Min Zhang
Let $\{M_n\}_{n=1}^{\infty}$ be a sequence of expanding positive integral matrices with $M_n= \begin{pmatrix} p_n & 0\\0 & q_n \end{pmatrix}$ for each $n\ge 1$, and let $D=\left\{\begin{pmatrix} 0\ 0 \end{pmatrix}, \begin{pmatrix} 1\ 0 \end{pmatrix}, \begin{pmatrix} 0\ 1 \end{pmatrix} \right\}$ be a finite digit set in $\mathbb{Z}^2$. The associated Borel probability measure obtained by an infinite convolution of atomic measures \[ \mu_{\{M_n\},D}=\delta_{M_1^{-1}D}*\delta_{(M_2M_1)^{-1}D}*\cdots*\delta_{(M_n\cdots M_2M_1)^{-1}D}*\cdots \] is called a Moran–Sierpinski-type measure. We prove that, under certain conditions, $\mu_{\{M_n\}, D}$ is a spectral measure if and only if $3\mid p_n$ and $3\mid q_n$ for each $n\geq2$.
65–80
Using digraphs to compute determinant, permanent, and Drazin inverse of circulant matrices with two parameters. Andrés M. Encinas, Daniel A. Jaume, Cristian Panelo, and Denis E. Videla
This work presents closed formulas for the determinant, permanent, inverse, and Drazin inverse of circulant matrices with two non-zero coefficients.
81–106
Existence and multiplicity of solutions for $p$-Kirchhoff-type Neumann problems. Qin Jiang, Sheng Ma, and Daniel Paşca
We establish, based on variational methods, existence theorems for a $p$-Kirchhoff-type Neumann problem under the Landesman–Lazer type condition and under the local coercive condition. In addition, multiple solutions for a $p$-Kirchhoff-type Neumann problem are established using a known three-critical-point theorem proposed by H. Brezis and L. Nirenberg.
107–121
Poincaré duality for Hopf algebroids. Sophie Chemla
We prove a twisted Poincaré duality for (full) Hopf algebroids with bijective antipode. As an application, we recover the Hochschild twisted Poincaré duality of van den Bergh. We also get a Poisson twisted Poincaré duality, which was already stated for oriented Poisson manifolds by Chen et al.
123–136
Gorenstein properties of split-by-nilpotent extension algebras. Pamela Suarez
Let $A$ be a finite-dimensional $k$-algebra over an algebraically closed field $k$. In this note, we study the Gorenstein homological properties of a split-by-nilpotent extension algebra. Let $R$ be a split-by-nilpotent extension of $A$. We provide sufficient conditions to ensure when a Gorenstein-projective module over $A$ induces a similar structure over $R$. We also study when a Gorenstein-projective $R$-module induces a Gorenstein-projective $A$-module. Moreover, we study the relationship between the Gorensteinness of $A$ and $R$.
137–144
Distance Laplacian eigenvalues of graphs, and chromatic and independence number. Shariefuddin Pirzada and Saleem Khan
Given an interval $I$, let $m_{D^{L} (G)} I$ (or simply $m_{D^{L}} I$) be the number of distance Laplacian eigenvalues of a graph $G$ which lie in $I$. For a prescribed interval $I$, we give the bounds for $m_{D^{L}}I$ in terms of the independence number $\alpha(G)$, the chromatic number $\chi$, the number of pendant vertices $p$, the number of components in the complement graph $C_{\overline{G}}$ and the diameter $d$ of $G$. In particular, we prove that $m_{D^{L}(G) }[n,n+2)\leq \chi-1$, $m_{D^{L}(G)}[n,n+\alpha(G))\leq n-\alpha(G)$, $m_{D^{L}(G) }[n,n+p)\leq n-p$ and discuss the cases where the bounds are best possible. In addition, we characterize graphs of diameter $d\leq 2$ which satisfy $m_{D^{L}(G) } (2n-1,2n )= \alpha(G)-1=\frac{n}{2}-1$. We also propose some problems of interest.
145–159
Limit behaviors for a $\beta$-mixing sequence in the St. Petersburg game. Yu Miao, Qing Yin, and Zhen Wang
We consider a sequence of non-negative $\beta$-mixing random variables $\{X,X_n : n\geq1\}$ from the classical St. Petersburg game. The accumulated gains $S_n=X_1+X_2+\cdots+X_n$ in the St. Petersburg game are studied, and the large deviations and the weak law of large numbers of $S_n$ are obtained.
161–171
Trivial extensions of monomial algebras. María Andrea Gatica, María Valeria Hernández, and María Inés Platzeck
We describe the ideal of relations for the trivial extension $T(\Lambda)$ of a finite-dimensional monomial algebra $\Lambda$. When $\Lambda$ is, moreover, a gentle algebra, we solve the converse problem: given an algebra $B$, determine whether $B$ is the trivial extension of a gentle algebra. We characterize such algebras $B$ through properties of the cycles of their quiver, and show how to obtain all gentle algebras $\Lambda$ such that $T(\Lambda) \cong B$. We prove that indecomposable trivial extensions of gentle algebras coincide with Brauer graph algebras with multiplicity one in all vertices in the associated Brauer graph, result proven by S. Schroll.
173–196
On the Eneström–Kakeya theorem and its various forms in the quaternionic setting. Abdullah Mir
We study the extensions of the classical Eneström–Kakeya theorem and its various generalizations regarding the distribution of zeros of polynomials from the complex to the quaternionic setting. We aim to build upon the previous work by various authors and derive zero-free regions of some special regular functions of a quaternionic variable with restricted coefficients, namely quaternionic coefficients whose real and imaginary components or moduli of the coefficients satisfy suitable inequalities. The obtained results for this subclass of polynomials and slice regular functions produce generalizations of a number of results known in the literature on this subject.
197–211
On the planarity, genus, and crosscap of the weakly zero-divisor graph of commutative rings. Nadeem ur Rehman, Mohd Nazim, and Shabir Ahmad Mir
Let $R$ be a commutative ring and $Z(R)$ its zero-divisors set. The weakly zero-divisor graph of $R$, denoted by $W\Gamma(R)$, is an undirected graph with the nonzero zero-divisors $Z(R)^*$ as vertex set and two distinct vertices $x$ and $y$ are adjacent if and only if there exist $a \in \mathrm{Ann}(x)$ and $b \in \mathrm{Ann}(y)$ such that $ab = 0$. In this paper, we characterize finite rings $R$ for which the weakly zero-divisor graph $W\Gamma(R)$ belongs to some well-known families of graphs. Further, we classify the finite rings $R$ for which $W\Gamma(R)$ is planar, toroidal or double toroidal. Finally, we classify the finite rings $R$ for which the graph $W\Gamma(R)$ has crosscap at most two.
213–227
Modular automata. Thomas N. Hibbard, Camilo A. Jadur, and Jorge F. Yazlle
Let $M$ and $b$ be integers greater than 1, and let $\mathbf{p}$ be a positive probability vector for the alphabet $\mathcal{A}_{b}=\{0,\ldots,b-1\}$. Let us consider a random sequence $w_0,w_1,\ldots,w_j$ over $\mathcal{A}_{b}$, where the $w_i$'s are independent and identically distributed according to $\mathbf{p}$. Such a sequence represents, in base $b$, the number $n=\sum_{i=0}^j w_i b^{j-i}$. In this paper, we explore the asymptotic distribution of $n\mbox{ mod }M$, the remainder of $n$ divided by $M$. In particular, by using the theory of Markov chains, we show that if $M$ and $b$ are coprime, then $n\mbox{ mod } M$ exhibits an asymptotic discrete uniform distribution, independent of $\mathbf{p}$; on the other hand, when $M$ and $b$ are not coprime, $n\mbox{ mod }M$ does not necessarily have a uniform distribution, and we obtain an explicit expression for this limiting distribution.
229–244
The principal small intersection graph of a commutative ring. Soheila Khojasteh
Let $R$ be a commutative ring with non-zero identity. The small intersection graph of $R$, denoted by $G(R)$, is a graph with the vertex set $V(G(R))$, where $V(G(R))$ is the set of all proper non-small ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if $I \cap J$ is not small in $R$. In this paper, we introduce a certain subgraph $PG(R)$ of $G(R)$, called the principal small intersection graph of $R$. It is the subgraph of $G(R)$ induced by the set of all proper principal non-small ideals of $R$. We study the diameter, the girth, the clique number, the independence number and the domination number of $PG(R)$. Moreover, we present some results on the complement of the principal small intersection graph.
245–256
Sequential optimality conditions for optimization problems with additional abstract set constraints. Nadia Soledad Fazzio, María Daniela Sánchez, and María Laura Schuverdt
The positive approximate Karush–Kuhn–Tucker sequential condition and the strict constraint qualification associated with this condition for general scalar problems with equality and inequality constraints have recently been introduced. In this paper, we extend them to optimization problems with additional abstract set constraints. We also present an extension of the approximate Karush–Kuhn–Tucker sequential condition and its related strict constraint qualification. Furthermore, we explore the relations between the new constraint qualification and other constraint qualifications known in the literature as Abadie, quasi-normality and the approximate Karush–Kuhn–Tucker regularity constraint qualification. Finally, we introduce an augmented Lagrangian method for solving the optimization problem with abstract set constraints and we show that it is possible to obtain global convergence under the new condition.
257–279
Properties of the convolution operation in the complexity space and its dual. José M. Hernández-Morales, Netzahualcóyotl C. Castañeda-Roldán, and Luz C. Álvarez-Marín
We give the basic properties of discrete convolution in the space of complexity functions and its dual space. Two inequalities are identified, and defined in the general context of an arbitrary binary operation in any weighted quasi-metric space. In that setting, some quasi-metric and convergence consequences of those inequalities are proven. Using convolution, we show a method for building improver functionals in the complexity space. We also consider convolution in three topologies within the dual space, obtaining two topological monoids.
281–299
Primitive decompositions of Dolbeault harmonic forms on compact almost-Kähler manifolds. Andrea Cattaneo, Nicoletta Tardini, and Adriano Tomassini
Let $(X,J,g,\omega)$ be a compact $2n$-dimensional almost-Kähler manifold. We prove primitive decompositions of $\partial$-, $\bar\partial$-harmonic forms on $X$ in bidegree $(1,1)$ and $(n-1,n-1)$ (such bidegrees appear to be optimal). We provide examples showing that in bidegree $(1,1)$ the $\partial$- and $\bar\partial$-decompositions differ.
301–316
Cofinite modules and cofiniteness of local cohomology modules. Alireza Vahidi, Ahmad Khaksari, and Mohammad Shirazipour
Let $n$ be a non-negative integer, $R$ a commutative Noetherian ring, $\mathfrak{a}$ an ideal of $R$, $M$ a finitely generated $R$-module, and $X$ an arbitrary $R$-module. In this paper, we first prove that if $\dim_R(M)\leq{n+2}$, then $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is an $(\operatorname{FD}_{ < n},\mathfrak{a})$-cofinite $R$-module and $\{\mathfrak{p}\in\operatorname{Ass}_R(\operatorname{H}^{i}_{\mathfrak{a}}(M)):\dim(R/\mathfrak{p})\geq{n}\}$ is a finite set for all $i$. As a consequence, it follows that $\operatorname{Ass}_R(\operatorname{H}^{i}_{\mathfrak{a}}(M))$ is a finite set for all $i$ when $R$ is a semi-local ring and $\dim_R(M)\leq{3}$. Then, we show that if $\dim(R/\mathfrak{a})\leq{n+1}$, then $\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},X)$ is an $\operatorname{FD}_{ < n}$ $R$-module for all $i$ whenever $\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},X)$ is an $\operatorname{FD}_{ < n}$ $R$-module for all $i\leq{\dim_R(X)-n}$. Finally, in the case that $\dim(R/\mathfrak{a})\leq{2}$, $X$ is $\mathfrak{a}$-torsion, and $n>0$ or $\operatorname{Supp}_R(X)\cap\operatorname{Var}(\mathfrak{a})\cap\operatorname{Max}(R)$ is finite, we prove that $X$ is an $(\operatorname{FD}_{ < n},\mathfrak{a})$-cofinite $R$-module when $\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},X)$ is an $\operatorname{FD}_{ < n}$ $R$-module for all $i\leq{2-n}$. We conclude with some ordinary $\mathfrak{a}$-cofiniteness results for local cohomology modules $\operatorname{H}^{i}_{\mathfrak{a}}(X)$.
317–325
Lorentz ${C}_{12}$-manifolds. Adel Delloum and Gherici Beldjilali
The object of the present paper is to study $C_{12}$-structures on a manifold with Lorentzian metric. We focus here on Lorentzian $C_{12}$-structures, emphasizing their relationship and analogies with respect to the Riemannian case. Several interesting results are obtained. Next, we study Ricci solitons in Lorentzian $C_{12}$-manifolds.
327–337
Two constructions of bialgebroids and their relations. Yudai Otsuto
We generalize the construction of face algebras by Hayashi and obtain a left bialgebroid $\mathfrak{A}(w)$. There are some relations between the left bialgebroid $\mathfrak{A}(w)$ and the generalized Shibukawa–Takeuchi left bialgebroid $A_{\sigma}$.
339–395

Volume 67, number 2 (2024)

December 2024
Front matter
Decidable objects and molecular toposes. Matías Menni
We study several sufficient conditions for the molecularity/local-connectedness of geometric morphisms. In particular, we show that if $\mathcal{S}$ is a Boolean topos, then, for every hyperconnected essential geometric morphism $p : \mathcal{E} \rightarrow \mathcal{S}$ such that the leftmost adjoint $p_{!}$ preserves finite products, $p$ is molecular and $p^* : \mathcal{S} \rightarrow \mathcal{E}$ coincides with the full subcategory of decidable objects in $\mathcal{E}$. We also characterize the reflections between categories with finite limits that induce molecular maps between the respective presheaf toposes. As a corollary we establish the molecularity of certain geometric morphisms between Gaeta toposes.
397–415
Extinction time of an epidemic with infection-age-dependent infectivity. Anicet Mougabe-Peurkor, Ibrahima Dramé, Modeste N'zi, and Étienne Pardoux
This paper studies the distribution function of the time of extinction of a subcritical epidemic, when a large enough proportion of the population has been immunized and/or the infectivity of the infectious individuals has been reduced, so that the effective reproduction number is less than one. We do that for a SIR/SEIR model, where infectious individuals have an infection-age-dependent infectivity, as in the model introduced in Kermack and McKendrick's seminal 1927 paper. Our main conclusion is that simplifying the model as an ODE SIR model, as it is largely done in the epidemics literature, introduces a bias toward shorter extinction time.
417–443
On hyponormality and a commuting property of Toeplitz operators. Houcine Sadraoui and Borhen Halouani
In this work we give sufficient conditions for hyponormality of Toeplitz operators on a weighted Bergman space when the analytic part of the symbol is a monomial and the conjugate part is a polynomial. We also extend a known commuting property of Toeplitz operators with a harmonic symbol on the Bergman space to weighted Bergman spaces.
445–453
Genus and book thickness of reduced cozero-divisor graphs of commutative rings. Edward Jesili, Krishnan Selvakumar, and Thirugnanam Tamizh Chelvam
For a commutative ring $R$ with identity, let $\langle a\rangle$ be the principal ideal generated by $a\in R$. Let $\Omega(R)^*$ be the set of all nonzero proper principal ideals of $R$. The reduced cozero-divisor graph $\Gamma_r(R)$ of $R$ is the simple undirected graph whose vertex set is $\Omega(R)^*$ and such that two distinct vertices $\langle a\rangle$ and $\langle b\rangle$ in $\Omega(R)^\ast$ are adjacent if and only if $\langle a \rangle\nsubseteq\langle b\rangle$ and $\langle b\rangle\nsubseteq\langle a\rangle$. In this article, we study certain properties of embeddings of the reduced cozero-divisor graph of commutative rings. More specifically, we characterize all Artinian nonlocal rings whose reduced cozero-divisor graph has genus two. Also we find the book thickness of the reduced cozero-divisor graphs which have genus at most one.
455–473
Boundedness of geometric invariants near a singularity which is a suspension of a singular curve. Luciana F. Martins, Kentaro Saji, Samuel P. dos Santos, and Keisuke Teramoto
Near a singular point of a surface or a curve, geometric invariants diverge in general, and the orders of this divergence, in particular the boundedness about these invariants, represent the geometry of the surface and the curve. In this paper, we study the boundedness and orders of several geometric invariants near a singular point of a surface which is a suspension of a singular curve in the plane, and those of the curves passing through the singular point. We evaluate the orders of the Gaussian and mean curvatures, as well as those of the geodesic and normal curvatures, and the geodesic torsion for the curve.
475–502
Complete presentation and Hilbert series of the mixed braid monoid $MB_{1,3}$. Zaffar Iqbal, Muhammad Mobeen Munir, Maleeha Ayub, and Abdul Rauf Nizami
The Hilbert series is the simplest way of finding dimension and degree of an algebraic variety defined explicitly by polynomial equations. The mixed braid groups were introduced by Sofia Lambropoulou in 2000. In this paper we compute the complete presentation and the Hilbert series of the canonical words of the mixed braid monoid $MB_{1,3}$.
503–516
One-sided EP elements in rings with involution. Cang Wu, Jianlong Chen, and Yu Chen
This paper investigates the one-sided EP property of elements in rings with involution. Let $R$ be a ring with involution $\ast$. Then $a \in R$ is said to be left (resp. right) EP if $a$ is Moore–Penrose invertible and $aR \subseteq a^{\ast}R$ (resp. $a^{\ast}R \subseteq aR$). Many properties of EP elements are extended to one-sided versions. Some new characterizations of EP elements are presented in relation to the absorption law for Moore–Penrose inverses.
517–528
Evolution of first eigenvalues of some geometric operators under the rescaled List's extended Ricci flow. Shahroud Azami and Abimbola Abolarinwa
Let $(M,g(t), e^{-\phi}d\nu)$ be a compact weighted Riemannian manifold and let $(g(t),\phi(t))$ evolve by the rescaled List's extended Ricci flow. In this paper, we study the evolution equations for first eigenvalues of the geometric operators, $-\Delta_{\phi}+cS^{a}$, along the rescaled List's extended Ricci flow. Here $\Delta_{\phi}=\Delta-\nabla\phi.\nabla$ is a symmetric diffusion operator, $\phi\in C^{\infty}(M)$, $S=R-\alpha|\nabla \phi|^{2}$, $R$ is the scalar curvature with respect to the metric $g(t)$ and $a, c$ are some constants. As an application, we obtain some monotonicity results under the rescaled List's extended Ricci flow.
529–543
The $w$-core–EP inverse in rings with involution. Dijana Mosić, Huihui Zhu, and Liyun Wu
The main goal of this paper is to present two new classes of generalized inverses in order to extend the concepts of the (dual) core–EP inverse and the (dual) $w$-core inverse. Precisely, we introduce the $w$-core–EP inverse and its dual for elements of a ring with involution. We characterize the (dual) $w$-core–EP invertible elements and develop several representations of the $w$-core–EP inverse and its dual in terms of different well-known generalized inverses. Using these results, we get new characterizations and expressions for the core–EP inverse and its dual. We apply the dual $w$-core–EP inverse to solve certain operator equations and give their general solution forms.
545–565
The limit case in a nonlocal $p$-Laplacian equation with dynamical boundary conditions. Eylem Öztürk
In this paper we deal with the limit as $p\to \infty$ for the nonlocal analogous to the $p$-Laplacian evolution with dynamic boundary conditions. Our main result demonstrates this limit in both the elliptic and parabolic cases. We are interested in smooth and singular kernels and show the existence and uniqueness of a limit solution. We obtain that the limit solution of the elliptic problem turns out to be also a viscosity solution of a corresponding problem. We prove that the natural energy functionals associated with this problem converge, in the sense of Mosco, to a limit functional and therefore we obtain convergence of solutions to the evolution problems in the parabolic case. For the limit problem, we provide examples of explicit solutions for some particular data.
567–591